arXiv subjects

Benson Farb

Publications and source records attributed to Benson Farb.

At least 19 recordsLinked to original sources

arXiv2026

Extremal mappings of tori, Teichm\"uller potentials and symmetric-space distance

The symmetric space $X_n={\rm SL}(n,\Rb)/{\rm SO}(n)$ can be interpreted as the Teichm\"uller space of marked, unit volume, flat $n$-dimensional tori. It comes with a unique (up to scale) ${\rm SL}(n,\Rb)$-invariant metric $d_{X_n}$. In 1939 Teichm\"uller gave a modular interpretation of $d_{X_2}$ (the hyperbolic metric) in terms of an extremal mapping problem for quasiconformal dilatation. Such a modular interpretation for $d_{X_n}$ for $n\geq 3$ has remained unaddressed: the natural candidates - minimal quasiconformal dilatation, Lipschitz constant, or total energy - do not work. In this paper we give such a modular interpretation, two in fact. We introduce the {\em total expansion} $\TE(f)\in [0,\infty]$ of a Lipschitz map $f:M\to N$ between Riemannian manifolds, a notion related to the notion of ``$k$-dilatation'' developed by Gromov, Guth and others. For volume-preserving Lipschitz maps $f:\Tc_0\to\Tc_1$ between $n$-dimensional, flat, unit-volume tori, we prove that $\TE(f)$ is minimized in the homotopy class of $f$ precisely by the affine maps in that class and takes on these the value $d_{X_n}$. We prove similar results for the \emph{Hilbert-Schmidt expansion} $\HE(f)$, which is a simple integral over $M$ and has more of an $L^2$ flavor.

math.DG↗

arXiv2026

Automorphisms of the moduli space of smooth cubic surfaces and its fundamental group

Let $\mathcal{C}$ be the moduli space of smooth complex cubic surfaces and let $\pi_1(\mathcal{C})$ be its (orbifold) fundamental group. We prove that the ``divisor subgroup'' of $\pi_1(\mathcal{C})$ is characteristic. This can be interpreted as saying that the group theory of $\pi_1(\mathcal{C})$ ``remembers'' the divisor of nodal cubic surfaces. We deduce from this group-theoretic result and some basic complex analysis that $\mathcal{C}$ has no nontrivial biholomorphic automorphisms as complex analytic orbifold.

math.AG↗

arXiv2025

Essential dimension relative to branched covers of degree at most n

We prove for various finite groups $G$ and integers $n\geq 1$ that there are families of equations with Galois group $G$ that cannot be simplified to a one-parameter family even after adjoining a root of a polynomial of degree at most $n$. In more geometric language, there are $G$-varieties $X$ with the following property: for any $G$-equivariant branched cover $\widetilde{X}\to X$ of degree $\leq n$, there is no dominant rational $G$-map $\widetilde{X}\dashrightarrow C$ to any $G$-curve $C$. The method of proof is new, and applies in cases where previous methods do not.

math.AG↗

arXiv2025

Entropy-minimizing diffeomorphisms of pseudo-Anosov type on K3 surfaces

We construct diffeomorphisms of ``pseudo-Anosov type'' on K3 surfaces M. In particular we obtain infinitely many examples of such diffeomorphisms that minimize entropy in their homotopy class, and for which neither the diffeomorphism nor any diffeomorphism homotopic to it preserves any complex structure on M.

math.DS↗

arXiv2024

Irrationality of the general smooth quartic $3$-fold using intermediate Jacobians

We prove that the intermediate Jacobian of the Klein quartic $3$-fold $X$ is not isomorphic, as a principally polarized abelian variety, to a product of Jacobians of curves. As corollaries we deduce (using a criterion of Clemens-Griffiths) that $X$, as well as the general smooth quartic $3$-fold, is irrational. These corollaries were known: Iskovskih-Manin \cite{IM} proved that every smooth quartic $3$-fold is irrational. However, the method of proof here is different than that of \cite{IM} and is significantly simpler.

math.AG↗

arXiv2024

The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

This is a paper in smooth $4$-manifold topology, inspired by the N\'{e}ron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(\pi)$ of Mordell-Weil group of an elliptic fibration $\pi:M\to\Pb^1$ is finitely generated. We compute $\MW(\pi_d)$ explicitly for elliptic fibrations $\pi_d:M_d\to\Pb^1$, where $M_d$ is a simply-connected complex surfaces $M_d$ of arithmetic genus $d\geq 1$ and all fibers of $\pi_d$ are nodal. We prove in this case that the fibered structure is unique up topological isotopy. By combining this with a result of Donaldson, we obtain the following remarkable consequence: any diffeomorphism of $M_d$ with $d\geq 3$ is topologically isotopic to a diffeomorphism taking fibers to fibers.

math.GT↗

arXiv2023

Moduli spaces and period mappings of genus one fibered K3 surfaces

In this paper we construct various moduli spaces of K3 surfaces $M$ equipped with a surjective holomorphic map $\pi:M\to\Pb^1$ with generic fiber a complex torus (e.g., an elliptic fibration). Examples include moduli spaces of such maps with primitive fibers; with reduced, irreducible fibers; equipped with a section; etc. Such spaces are closely related to the moduli space of Ricci-flat metrics on $M$. We construct period mappings relating these moduli spaces to locally symmetric spaces, and use these to compute their (orbifold) fundamental groups. These results lie in contrast to, and exhibit different behavior than, the well-studied case of moduli spaces of polarized K3 surfaces, and are more useful for applications to the mapping class group $\Mod(M)$. Indeed, we apply our results on moduli space to give two applications to the smooth mapping class group of $M$.

math.AG↗

arXiv2023

Rigidity of moduli spaces and algebro-geometric constructions

In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and holomorphic aspects, and so they also provide motivation for natural problems in geometric group theory and topology.

math.AG↗

arXiv2022

Global rigidity of the period mapping

Let ${\mathcal M}_{g,n}$ denote the moduli space of smooth, genus $g\geq 1$ curves with $n\geq 0$ marked points. Let ${\mathcal A}_h$ denote the moduli space of $h$-dimensional, principally polarized abelian varieties. Let $g\geq 3$ and $h\leq g$. If $F:{\mathcal M}_{g,n}\to{\mathcal A}_h$ is a nonconstant holomorphic map then $h=g$ and $F$ is the classical period mapping, assigning to a Riemann surface $X$ its Jacobian.

math.AG↗

arXiv2022

The Nielsen realization problem for K3 surfaces

The smooth (resp. metric and complex) Nielsen Realization Problem for K3 surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-flat metric, or automorphisms of a complex structure)? We solve the metric and complex versions of Nielsen Realization, and we solve the smooth version almost completely for involutions. Unlike the case of $2$-manifolds, some $G$ are realizable and some are not, and the answer depends on the category of structure preserved. In particular, Dehn twists are not realizable by finite order diffeomorphisms. We introduce a computable invariant $L_G$ that determines in many cases whether $G$ is realizable or not, and apply this invariant to construct an $S_4$ action by isometries of some Ricci-flat metric on $M$ that preserves no complex structure. We also show that the subgroups of ${\rm Diff}(M)$ of a given prime order $p$ which fix pointwise some positive-definite $3$-plane in $H_2(M;\mathbb{R})$ and preserve some complex structure on $M$ form a single conjugacy class in ${\rm Diff}(M)$ (it is known that then $p\in \{2,3,5,7\}$).

math.GT↗

arXiv2021

Essential dimension via prismatic cohomology

For $X$ a smooth, proper complex variety we show that for $p\gg 0$, the restriction of the mod $p$ cohomology $H^i(X,\mathbb{F}_p)$ to any Zariski open has dimension at least $h^{0,i}_X$. The proof uses the prismatic cohomology of Bhatt-Scholze. We use this result to obtain lower bounds on the $p$-essential dimension of covers of complex varieties. For example, we prove the $p$-incompressibility of the mod $p$ homology cover of an abelian variety, confirming a conjecture of Brosnan for sufficiently large $p.$ By combining these techniques with the theory of toroidal compactifications of Shimura varieties, we show that for any Hermitian symmetric domain $X,$ there exist $p$-congruence covers that are $p$-incompressible.

math.AG↗

arXiv2020

Irreducible Sp-representations and subgroup distortion in the mapping class group

We prove that various subgroups of the mapping class group $Mod(Σ)$ of a surface $Σ$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstadt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of $H_1(Σ;\mathbb{Z})$.

math.GT↗

arXiv2020

The Essential Dimension of Congruence Covers

Consider the algebraic function $Φ_{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Kronecker and Klein asks: What is the minimal $d$ such that, after a rational change of variables, the function $Φ_{g,n}$ can be written as an algebraic function of $d$ variables? Using techniques from the deformation theory of $p$-divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $p$-dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $p$-dimension of congruence covers of the moduli space of genus $g$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties.

math.AG↗

arXiv2020

Resolvent degree, Hilbert's 13th Problem and geometry

We develop the theory of resolvent degree, introduced by Brauer \cite{Br} in order to study the complexity of formulas for roots of polynomials and to give a precise formulation of Hilbert's 13th Problem. We extend the context of this theory to enumerative problems in algebraic geometry, and consider it as an intrinsic invariant of a finite group. As one application of this point of view, we prove that Hilbert's 13th Problem, and his Sextic and Octic Conjectures, are equivalent to various enumerative geometry problems, for example problems of finding lines on a smooth cubic surface or bitangents on a smooth planar quartic.

math.AG↗

arXiv2019

Modular functions and resolvent problems

The link between modular functions and algebraic functions was a driving force behind the 19th century study of both. Examples include the solutions by Hermite and Klein of the quintic via elliptic modular functions and the general sextic via level $2$ hyperelliptic functions. This paper aims to apply modern arithmetic techniques to the circle of ``resolvent problems'' formulated and pursued by Klein, Hilbert and others. As one example, we prove that the essential dimension at $p=2$ for the symmetric groups $S_n$ is equal to the essential dimension at $2$ of certain $S_n$-coverings defined using moduli spaces of principally polarized abelian varieties. Our proofs use the deformation theory of abelian varieties in characteristic $p$, specifically Serre-Tate theory, as well as a family of remarkable mod $2$ symplectic $S_n$-representations constructed by Jordan. As shown in an appendix by Nate Harman, the properties we need for such representations exist only in the $p=2$ case. In the second half of this paper we introduce the notion of $\E$-versality as a kind of generalization of Kummer theory, and we prove that many congruence covers are $\E$-versal. We use these $\E$-versality result to deduce the equivalence of Hilbert's 13th Problem (and related conjectures) with problems about congruence covers.

math.AG↗

arXiv2019

Geometry of the Wiman-Edge pencil and the Wiman curve

The {\em Wiman-Edge pencil} is the universal family $C_t, t\in\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The curve $C_0$, discovered by Wiman in 1895 \cite{Wiman} and called the {\em Wiman curve}, is the unique smooth, genus $6$ curve admitting a faithful action of the symmetric group $\Sf_5$. In this paper we give an explicit uniformization of $\mathcal B$ as a non-congruence quotient $Γ\backslash \Hf$ of the hyperbolic plane $\Hf$, where $Γ<\PSL_2(\Z)$ is a subgroup of index $18$. We also give modular interpretations for various aspects of this uniformization, for example for the degenerations of $C_t$ into $10$ lines (resp.\ $5$ conics) whose intersection graph is the Petersen graph (resp.\ $K_5$). In the second half of this paper we give an explicit arithmetic uniformization of the Wiman curve $C_0$ itself as the quotient $Λ\backslash \Hf$, where $Λ$ is a principal level $5$ subgroup of a certain "unit spinor norm" group of Möbius transformations. We then prove that $C_0$ is a certain moduli space of Hodge structures, endowing it with the structure of a Shimura curve of indefinite quaternionic type.

math.AG↗

arXiv2019

Arithmeticity of the monodromy of the Wiman-Edge pencil

The {\em Wiman-Edge pencil} is the universal family $\Cs/\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The goal of this paper is to prove that the monodromy of $\Cs/\mathcal B$ is commensurable with a Hilbert modular group; in particular is arithmetic. We then give a modular interpretation of this, as well as a uniformization of $\mathcal B$.

math.AG↗

arXiv2019

Coincidences of homological densities, predicted by arithmetic

Motivated by analogies with basic density theorems in analytic number theory, we introduce a notion (and variations) of the homological density of one space in another. We use Weil's number field/ function field analogy to predict coincidences for limiting homological densities of various sequences $\mathcal{Z}^{(d_1,\ldots,d_m)}_n(X)$ of spaces of $0$-cycles on manifolds $X$. The main theorem in this paper is that these topological predictions, which seem strange from a purely topological viewpoint, are indeed true. The obstacle to proving such a theorem with current technology is how to deal with the combinatorial complexity of all possible "collisions" of points, this problem does not arise in the simplest (and classical) case $(m,n)=(1,2)$ of configuration spaces. To overcome this obstacle we develop a method that uses the Björner--Wachs theory of lexicographic shellability from algebraic combinatorics to study such problems. As a consequence we derive new homological stability theorems for broad classes of $0$-cycles on manifolds. Even in the classical case $(m,n)=(1,2)$ this gives a new, simplified proof of classical results, and also of recent theorems of Church and others.

math.AT↗